Similarly to this thread NSolve gives additional solutions that don't satisfy the equations!
NSolve returns "spurious" solutions, even increasing the working precision
fsys={1 - Cos[(11 π)/45] - (3 Cos[ψ1])/4 - 2/3 Sin[φ + π/6] == 0, h + 2/3 Cos[φ + π/6] + Sin[(11 π)/45] - (3 Sin[ψ1])/4 == 0, -0.0436952 - (3 Cos[ψ2])/4 - 2/3 Sin[φ - π/6] == 0,2/5 + h + 2/3 Cos[φ - π/6] + Sin[(17 π)/180] - (3 Sin[ψ2])/4 == 0}
NSolve[N[fsys], {φ, h, ψ1, ψ2}, WorkingPrecision -> 100]
The second solution returned is
{h -> -1.99515, ψ2 -> -1.22627, ψ1 -> 1.68803, φ -> 0.0618559,
Sin[ψ1] -> -0.993136, Cos[ψ1] -> -0.116967,
Sin[ψ2] -> -0.941237, Cos[ψ2] -> 0.337748, Sin[φ] -> 0.0618165,
Cos[φ] -> 0.998088}, {h -> -1.99515, ψ2 -> 1.22627,
ψ1 -> -1.68803, φ -> 0.0618559, Sin[ψ1] -> -0.993136,
Cos[ψ1] -> -0.116967, Sin[ψ2] -> -0.941237,
Cos[ψ2] -> 0.337748, Sin[φ] -> 0.0618165, Cos[φ] -> 0.998088}
,
I do not understand why both ψ1 and Cos[ψ1] are returned, with Sin[1.68803] actually not equal to 0.993136 (the sign is different)
Sin[...], Cos[...]bits (on version 9). $\endgroup$NSolveis not the right tool as stated in the documentation and the warnings. Why not usingFindRoot? $\endgroup$